# Why do any impurities lower the melting point of an isolated substance?

It is known that impurities in a desired isolated product lower the melting point of the mixture, even if the impurities melting point is much higher than the desired product. Why is that?

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I think your question is too vague. F'x gave you a great answer on the subject of metal mixtures, but you could get one equally good on colligative properties of solutions, or on thermodynamics. What particular case are you thinking of? A synthesis product not pure enough? A solution of glucose in water? –  CHM Apr 26 '12 at 0:59
It's what a TA told me after my observation in the organic lab; indeed a synthesis product wasn't exactly pure. –  LanceLafontaine Apr 26 '12 at 1:12
Indeed this is a very common phenomenon, not limited to alloys. This is basis for use of mp as a kriterion for substance indentity and purity. There are exeptions (very few in organic chemisty) eg when the impurity and the substance make up a compound (salt). Nick T 's explanation by thermodynamics of mixed phases is the right one. –  Georg Apr 27 '12 at 9:18

It's a very general statement, but it's not always true. I'll explain why it's often true, and give a counter-example at the end.

Your majority component B and the impurity (let's call it A) form a binary system. In most cases, such binary mixtures exhibit a solid–liquid phase diagram as follows:

(image taken from these lecture notes).

This binary phase diagram has pure A on the left, pure B on the right. A and B form, somewhere, a eutectic. It is the point here at concentration e and temperature y. Because the existence of a eutectic point is guaranteed for any A/B binary system, and because the eutectic corresponds to a lower temperature, your liquidus curve decreases with increasing impurity concentration, and the impurity thus lowers the melting point.

However, not all binary mixtures form a eutectic. In the words of Wikipedia:

Not all binary alloys have a eutectic point; for example, in the silver-gold system the melt temperature (liquidus) and freeze temperature (solidus) both increase monotonically as the mix changes from pure silver to pure gold.

The corresponding phase diagram is as follows:

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Thermodynamically, you're considering the chemical potentials ($\mu$) of the liquid and solid(s), specifically the temperature where they're equal. In a mixture, the potential is lower as the disorder (entropy) has increased, so all things equal, it will favor the liquid over a purer solid where there can be more disorder. Chemical systems seek to lower their potential through spontaneous chemical changes (e.g. of phase), minimizing the free energy ($G$). Equilibrium occurs where $dG = 0$. In a constant temperature, constant pressure system,

$$dG =\sum_i\mu_idn_i$$

$\mu$ is the chemical potential of a given species (some compound in some phase) and $n$ is the amount of that compound.

Assuming our solid generated is pure (valid sometimes, not always), our reaction (freezing) is $$\require{mhchem} \ce{A_{(l)} -> A_{(s)}}$$

so the relationship between the two species is equal and opposite (generating one mole of solid consumes one mole of liquid). $$-dn_{(l)} = dn_{(s)}$$

therefore,

\begin{align} dG = 0 &= \mu{(l)}dn_{(l)} + \mu{(s)}dn_{(s)}\\ &= \mu{(l)}dn_{(l)} - \mu{(s)}dn_{(l)}\\ 0 &= \mu{(l)} - \mu{(s)}\\ \mu{(s)} &= \mu{(l)}\\ \end{align}

As the solid is pure (prior assumption, note: $\star$ denotes pure compound),

$$\mu{(s)} = \mu^\star{(s)}$$

In an ideal mixture (assumes interactions between all components are equal) with mole fraction $\chi_A$ and $T$ being the freezing point of the mixture,

$$\mu{(l)} = \mu^\star{(l)} + RT \ln \chi_A$$

(If non-ideal, a general $a$ term is used $\mu{(l)} = \mu^\star{(l)} + RT \ln a_A$). All together:

\begin{align} \mu^\star{(s)} &= \mu{(s)} = \mu{(l)} = \mu^\star{(l)} + RT \ln \chi_A\\ \mu^\star{(s)} - \mu^\star{(l)} &= RT \ln \chi_A\\ -\Delta G^\star_{m, fus} &= RT \ln \chi_A\\ -(\Delta H^\star_{m, fus} - T \Delta S^\star_{m, fus}) &= RT \ln \chi_A\\ \frac{\Delta S^\star_{m, fus}}{R} - \frac{\Delta H^\star_{m, fus}}{RT} &= \ln \chi_A\\ \end{align}

If the mixture is pure (freezing point at $T^\star$), $\chi_A = 1$, so $\ln \chi_A = 0$,

\begin{align} -\frac{\Delta H^\star_{m, fus}}{RT^\star} + \frac{\Delta S^\star_{m, fus}}{R} &= 0\\ \frac{\Delta S^\star_{m, fus}}{R} &= \frac{\Delta H^\star_{m, fus}}{RT^\star} \\ \end{align}

Combined...

\begin{align} \frac{\Delta H^\star_{m, fus}}{RT^\star} - \frac{\Delta H^\star_{m, fus}}{RT} &= \ln \chi_A\\ \frac{\Delta H^\star_{m, fus}}{R}\left(\frac{1}{T} - \frac{1}{T^\star}\right) &= \ln \chi_A\\ \frac{\Delta H^\star_{m, fus}}{R}\left(\frac{T^\star - T}{TT^\star}\right) &= \ln \chi_A\\ \end{align}

As $T \approx T^\star$ and if we define $\Delta T$ as the change in equilibrium temperature (melting point) from the pure substance,

\begin{align} \frac{\Delta H^\star_{m, fus}}{R}\left(\frac{\Delta T}{T^{\star2}}\right) &= \ln \chi_A\\ \end{align}

This can be taken further to derive the coefficient for the freezing point depression, but from here we can see that because $\Delta H$, all $T$'s, and $R$ are positive and $\ln \chi_A$ is guaranteed to be negative ($\chi_A$ must be less than zero), $\Delta T$ must be negative.

To recap the assumptions, this is an ideal mixture and the solid formed is pure. Addressing the first, if non-ideal, we can use the general $a_A$ instead of $\chi_A$. The mole fraction can never exceed 1, but I'm not sure about $a$; if it does exceed 1, then there would be a freezing point elevation.

As for the second, if the solid formed is a mixture (e.g. a metal alloy), the that throws another wrench in the works that I don't definitively know how to address. I believe it would then be based on the difference of interaction between the components in liquid phase versus solid phase as well as the relative concentration in each.

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+1 very detailed answer. Read this answer and mine, you can extend your discussion of idealities into a discussion of why some binary mixtures don't form eutectics… –  F'x Apr 26 '12 at 3:00
It really doesn't have anything to do with eutectics, although you are quite right that many systems don't have eutectics. The derivation above is quite sufficient to show that the addition of any impurity in small quantities will lower the melting point. Another reason is the entropy of mixing, which when one component amount is small is always positive. –  Paul J. Gans Dec 22 '12 at 2:48